*2.2. Proposed Governing Equations*

Assuming the above-mentioned Reiner-Rivlin fluid model, the proposed governing equations for continuity and momentum in the direction of *r*, *θ*, *z* read as

$$\frac{1}{r}D\_2(rv\_\tau) + \frac{1}{r}D\_3(v\_\theta) + D\_4(v\_\overline{z}) = 0,\tag{14}$$

$$\begin{array}{l} \rho \left( D\_1(\upsilon\_r) + \upsilon\_r D\_2(\upsilon\_r) + \frac{\upsilon\_\theta}{r} D\_3(\upsilon\_r) + \upsilon\_z D\_4(\upsilon\_r) - \frac{\upsilon\_\theta}{r} \right) = -D\_2 p\\ \quad + \mu \left[ \frac{1}{r} D\_2(\upsilon\_r) + \frac{1}{r^2} D\_3^{-2}(\upsilon\_r) + D\_4^{-2}(\upsilon\_r) - \frac{2}{r} D\_3(\upsilon\_\theta) - \frac{\upsilon\_r}{r^2} \right] \\ \quad + \frac{1}{r} \frac{\partial}{\partial r}(r \tau\_{\ell\tau}) + \frac{1}{r} D\_3(\tau\_{r\theta}) - \frac{\tau\_{\theta\theta}}{r} + D\_4(\tau\_{rz}) - D\_4(B\_r) B\_z - D\_4(B\_\theta) B\_\theta \end{array} \tag{15}$$

$$\begin{array}{l} \rho \left( D\_1(\upsilon\_\theta) + \upsilon\_r D\_2(\upsilon\_\theta) + \frac{\upsilon\_\theta}{r} D\_3(\upsilon\_\theta) + \upsilon\_z D\_4(\upsilon\_\theta) - \frac{\upsilon\_r \upsilon\_\theta}{r} \right) = -\frac{1}{r} D\_3 p\\ + \mu \left[ \frac{1}{r} D\_2(r D\_2(\upsilon\_\theta)) + \frac{1}{r^2} D\_3^{-2}(\upsilon\_\theta) + D\_4^{-2}(\upsilon\_\theta) + \frac{2}{r^2} D\_3(\upsilon\_r) - \frac{\upsilon\_\theta}{r^2} \right] \\ + \frac{1}{r} D\_3(\tau\_\theta \rho) + \frac{1}{r^2} D\_2(r^2 \tau\_\theta \rho) + D\_4(\tau\_\theta z) - D\_4(B\_\theta) B\_2 - D\_2(B\_\theta) B\_r \end{array} \tag{16}$$

$$\begin{array}{c} \rho \left( D\_1(\upsilon\_z) + \upsilon\_r D\_2(\upsilon\_z) + \frac{\upsilon\_\theta}{r} D\_3(\upsilon\_z) + \upsilon\_z D\_4(\upsilon\_z) \right) = -D\_4 p\\ \quad + \mu \left[ \frac{1}{r} D\_2(r D\_2(\upsilon\_z)) + \frac{1}{r^2} D\_3^{-2}(\upsilon\_z) + D\_4^{-2}(\upsilon\_z) \right] \\ \quad + D\_4(\tau\_{zz}) + \frac{1}{r} D\_2(r \tau\_{rz}) + \frac{1}{r} D\_3(\tau\_{\theta z}) - D\_4(B\_\theta) B\_\theta + D\_4(B\_r) B\_r \end{array} \tag{17}$$

where *p* represents pressure, *ρ* represents fluid density, stress tensor is denoted by *τ*, and *μ* represents fluid viscosity. The equation of the magnetic field is

$$\frac{1}{r}D\_2rB\_I + \frac{1}{r}D\_3B\_\theta + D\_4B\_z = 0,\tag{18}$$

$$D\_1 B\_r + \upsilon\_r D\_2 B\_r + \upsilon\_\theta D\_3 B\_r + \upsilon\_z D\_4 B\_r = -D\_4 (\upsilon\_r B\_z - \upsilon\_z B\_r) + \frac{1}{\delta \mu\_2} \left( D\_4^{-2} B\_r \right),\tag{19}$$

$$\begin{aligned} D\_1 B\_\theta + \upsilon\_r D\_2 B\_\theta + \upsilon\_\theta D\_3 B\_\theta + \upsilon\_z D\_4 B\_\theta &= D\_2 (\upsilon\_r B\_\theta - \upsilon\_\theta B\_r) \\ -D\_4 (\upsilon\_\theta B\_z - B\_\theta \upsilon\_z) + \frac{1}{\delta \overline{\mu\_2}} \left( D\_4 \,^2 B\_\theta \right) \end{aligned} \tag{20}$$

$$D\_1 B\_z + v\_r D\_2 B\_z + v\_\theta D\_3 B\_z + v\_z D\_4 B\_z = D\_2 (v\_r B\_z - v\_z B\_r) + \frac{1}{\delta \mu\_2} \left( D\_4^{-2} B\_z \right),\tag{21}$$

where *δ* is the electrical conductivity. The energy equation reads as:

$$\begin{split} D\_1 \overset{\smile}{T} + \upsilon\_r D\_2 \overset{\smile}{T} + \upsilon\_z D\_4 \overset{\smile}{T} &= \frac{k}{(\rho c)\_f} D\_4 \overset{\smile}{T} - \frac{1}{(\rho c)\_f} \left( \frac{\partial q\_r}{\partial r} \right) \\ &+ \frac{(\rho c)\_p}{(\rho c)\_f} \left[ D\_B \left( D\_2 \overset{\smile}{T} \cdot D\_2 \overset{\smile}{\complement} + D\_4 \overset{\smile}{T} \cdot D\_4 \overset{\smile}{\complement} \right) + \frac{D\_T}{\bar{T}\_a} \left[ \left( D\_2 \overset{\smile}{T} \right)^2 + \left( D\_4 \overset{\smile}{T} \right)^2 \right] \right] \end{split} \tag{22}$$

where *T* represents temperature, *k* the thermal conductivity, *C* represents concentration, mean fluid temperature is represented by *Tm*, the specific heat capacity of nanofluid (*ρc*)*p*,

(*ρc*)*f* the specific heat capacity of the base fluid, Brownian diffusivity is represented by *DB*, thermophoretic diffusion coefficient is represented by *DT*. In accordance with Rosseland approximation radiation heat flux, which is uni-directional (acting axially) takes the form, *qr* = − 4*σe* 3*βr ∂ T* 4 *∂r* , in which *σe* represents the Stefan–Boltzmann constant and *βr* represents the mean absorption coefficient, respectively. Rosseland's model applies for optically thick nanofluids and yields a reasonable estimate for radiative transfer effects, although it neglectsnon-grayeffects.

The equation of nanoparticle concentration reads as [54]

$$D\_1 \overset{\smile}{\mathbb{C}} + v\_r D\_2 \overset{\smile}{\mathbb{C}} + v\_z D\_4 \overset{\smile}{\mathbb{C}} = D\_B D\_4 \overset{\smile}{\mathbb{C}} + \underbrace{\underset{\smile}{\underset{T\_u}{\smile}} D\_4^2 \overset{\smile}{T}}\_{T\_u} - k\_r^2 \left(\overset{\smile}{\mathbb{C}} - \overset{\smile}{\mathbb{C}}\_u\right) \left(\overset{\smile}{\underset{T\_u}{\smile}}\right)^n \varepsilon^{-\frac{\underline{\varepsilon}\_0}{n\underline{T}}},\tag{23}$$

where *k*2*r* is the reaction rate, *n* is the rate constant, *κ* is the Boltzmann constant, and *Ea* is the activation energy.

The microorganism conservation equation reads as

$$D\_1 n + v\_l D\_2 n + v\_\theta D\_3 n + v\_z D\_4 n + \frac{b \mathcal{W}\_{mo}}{\breve{C}\_l - \breve{C}\_u} \left[ D\_4 \left( n D\_4 \breve{C} \right) \right] = D\_{mo} \left( D\_4^{-2} n \right). \tag{24}$$

Here *bW*mo is considered constant, where *b* are chemotaxis constant, cell swimming maximal speed is denoted by *W*mo, and *D*mo denotes diffusivity of microorganism. The corresponding boundary conditions are [54].

$$\begin{aligned} \upsilon\_r = 0, \upsilon\_\theta = \Omega\_1 r \frac{D^2}{\stackrel{\frown 2}{\frown 2}}, \upsilon\_z = 0, B\_z = B\_\theta = 0, n = n\_l, \check{T} = \check{T}\_{l\prime} \check{\mathbb{C}} = \check{\mathbb{C}}\_{l\prime} \text{ at } z = 0, \\ \check{\mathbb{F}}\_\theta \text{ ( $t$ )} \end{aligned} \tag{25}$$

$$\begin{aligned} \upsilon\_r &= 0, \upsilon\_\theta = \Omega\_2 r \frac{D^2}{\Gamma^2}, B\_\theta = N\_0 r \frac{D^2}{\Gamma^2}, B\_z = -\frac{\beta DM\_0}{\Gamma}, \\\ \overset{\leftarrow}{C} &= \overset{\leftarrow}{C}\_{\mu\_\theta} \overset{\leftarrow}{T} = T\_{\mu\_\theta} n = n\_{\mu\_\theta} \upsilon\_\mathbb{z} = -\frac{\beta D^2}{2\Gamma(t)}, \end{aligned} \text{ \text{ at } z = \overset{\frown}{\Gamma}(t), \tag{26}$$
